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Properties

Label 880.2.co.a
Level $880$
Weight $2$
Character orbit 880.co
Analytic conductor $7.027$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(47,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 5, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.co (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{5} + 16 q^{13} + 12 q^{17} + 16 q^{21} - 4 q^{25} - 96 q^{33} + 12 q^{37} + 24 q^{41} - 48 q^{45} - 68 q^{53} - 48 q^{57} - 48 q^{61} - 128 q^{65} - 88 q^{73} + 40 q^{81} - 32 q^{85} + 12 q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.47972 + 2.90411i 0 −2.10334 − 0.758910i 0 1.29780 + 2.54707i 0 −4.48095 − 6.16750i 0
47.2 0 −1.36988 + 2.68854i 0 1.89460 − 1.18764i 0 −1.79957 − 3.53185i 0 −3.58833 − 4.93892i 0
47.3 0 −0.887637 + 1.74208i 0 0.484765 + 2.18289i 0 −0.0238627 − 0.0468332i 0 −0.483605 − 0.665625i 0
47.4 0 −0.609048 + 1.19532i 0 −2.22609 + 0.210994i 0 −0.965205 − 1.89432i 0 0.705496 + 0.971032i 0
47.5 0 −0.511174 + 1.00324i 0 2.15023 + 0.613619i 0 2.04147 + 4.00662i 0 1.01817 + 1.40139i 0
47.6 0 −0.356753 + 0.700167i 0 −0.112374 − 2.23324i 0 1.02329 + 2.00833i 0 1.40039 + 1.92748i 0
47.7 0 0.356753 − 0.700167i 0 −0.112374 − 2.23324i 0 −1.02329 − 2.00833i 0 1.40039 + 1.92748i 0
47.8 0 0.511174 − 1.00324i 0 2.15023 + 0.613619i 0 −2.04147 − 4.00662i 0 1.01817 + 1.40139i 0
47.9 0 0.609048 − 1.19532i 0 −2.22609 + 0.210994i 0 0.965205 + 1.89432i 0 0.705496 + 0.971032i 0
47.10 0 0.887637 − 1.74208i 0 0.484765 + 2.18289i 0 0.0238627 + 0.0468332i 0 −0.483605 − 0.665625i 0
47.11 0 1.36988 − 2.68854i 0 1.89460 − 1.18764i 0 1.79957 + 3.53185i 0 −3.58833 − 4.93892i 0
47.12 0 1.47972 − 2.90411i 0 −2.10334 − 0.758910i 0 −1.29780 − 2.54707i 0 −4.48095 − 6.16750i 0
207.1 0 −3.26197 − 0.516645i 0 0.204267 − 2.22672i 0 −2.05473 + 0.325437i 0 7.52034 + 2.44351i 0
207.2 0 −2.21075 − 0.350148i 0 −1.54554 + 1.61595i 0 −1.83409 + 0.290491i 0 1.91163 + 0.621127i 0
207.3 0 −2.07013 − 0.327876i 0 −2.22316 − 0.239952i 0 3.07033 − 0.486293i 0 1.32475 + 0.430438i 0
207.4 0 −1.22140 − 0.193450i 0 2.20016 − 0.399121i 0 −0.786807 + 0.124618i 0 −1.39878 − 0.454491i 0
207.5 0 −0.899291 − 0.142434i 0 0.459017 + 2.18845i 0 3.49542 − 0.553621i 0 −2.06473 − 0.670872i 0
207.6 0 −0.305063 − 0.0483172i 0 −0.545802 − 2.16843i 0 −3.07506 + 0.487042i 0 −2.76244 − 0.897571i 0
207.7 0 0.305063 + 0.0483172i 0 −0.545802 − 2.16843i 0 3.07506 − 0.487042i 0 −2.76244 − 0.897571i 0
207.8 0 0.899291 + 0.142434i 0 0.459017 + 2.18845i 0 −3.49542 + 0.553621i 0 −2.06473 − 0.670872i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
11.c even 5 1 inner
20.e even 4 1 inner
44.h odd 10 1 inner
55.k odd 20 1 inner
220.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.co.a ✓ 96
4.b odd 2 1 inner 880.2.co.a ✓ 96
5.c odd 4 1 inner 880.2.co.a ✓ 96
11.c even 5 1 inner 880.2.co.a ✓ 96
20.e even 4 1 inner 880.2.co.a ✓ 96
44.h odd 10 1 inner 880.2.co.a ✓ 96
55.k odd 20 1 inner 880.2.co.a ✓ 96
220.v even 20 1 inner 880.2.co.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.co.a ✓ 96 1.a even 1 1 trivial
880.2.co.a ✓ 96 4.b odd 2 1 inner
880.2.co.a ✓ 96 5.c odd 4 1 inner
880.2.co.a ✓ 96 11.c even 5 1 inner
880.2.co.a ✓ 96 20.e even 4 1 inner
880.2.co.a ✓ 96 44.h odd 10 1 inner
880.2.co.a ✓ 96 55.k odd 20 1 inner
880.2.co.a ✓ 96 220.v even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} - 136 T_{3}^{92} + 20610 T_{3}^{88} - 2697070 T_{3}^{84} + 332394795 T_{3}^{80} + \cdots + 52\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display